Chapter 13

A thin spherical conducting shell of radius \(R\) has a charge \(q\). Another charge \(Q\) is placed at the centre of the shell. The electrostatic potential at a point \(P\) at a distance \(R / 2\) from the centre of the shell is (A) \(\frac{2 Q}{4 \pi \varepsilon_{0} R}\) (B) \(\frac{2 Q}{4 \pi \varepsilon_{0} R}-\frac{2 q}{4 \pi \varepsilon_{0} R}\) (C) \(\frac{2 Q}{4 \pi \varepsilon_{0} R}+\frac{q}{4 \pi \varepsilon_{0} R}\) (D) \(\frac{(q+Q)}{4 \pi \varepsilon_{0}} \frac{2}{R}\)

A charged particle \(q\) is shot towards another charged particle \(Q\) which is fixed, with a speed \(v\). It approaches \(Q\) up to a closest distance \(\mathrm{r}\) and then returns. If \(q\) was given a speed \(2 v\), the closest distance of approach would be (A) \(r\) (B) \(2 r\) (C) \(r / 2\) (D) \(r / 4\)

Four charges equal to \(-Q\) are placed at the four corners of a square and a charge \(q\) is at its centre. If the system is in equilibrium, the value of \(q\) is (A) \(-\frac{Q}{4}(1+2 \sqrt{2})\) (B) \(\frac{Q}{4}(1+2 \sqrt{2})\) (C) \(-\frac{Q}{2}(1+2 \sqrt{2})\) (D) \(\frac{Q}{2}(1+2 \sqrt{2})\)

A charged ball \(B\) hangs from a silk thread \(S\), which makes an angle \(\theta\) with a large charged conducting sheet \(P\), as shown in Fig. 13.57. The surface charge density \(\sigma\) of the sheet is proportional to (A) \(\cos \theta\) (B) \(\cot \theta\) (C) \(\sin \theta\) (D) \(\tan \theta\)

Two point charges \(+8 q\) and \(-2 q\) are located at \(x=0\) and \(x=L\), respectively. The location of a point on the \(x\)-axis at which the net electric field due to these two point charges is zero (A) \(2 L\) (B) \(\frac{L}{4}\) (C) \(8 L\) (D) \(4 L\)

Two thin wire rings each having a radius \(R\) are placed at a distance \(d\) apart with their axes coinciding. The charges on the two rings are \(+q\) and \(-q .\) The potential difference between the centres of the two rings is (A) \(\frac{q R}{4 \pi \varepsilon_{0} d^{2}}\) (B) \(\frac{q}{2 \pi \varepsilon_{0}}\left[\frac{1}{R}-\frac{1}{\sqrt{R^{2}+d^{2}}}\right]\) (C) Zero (D) \(\frac{q}{4 \pi \varepsilon_{0}}\left[\frac{1}{R}-\frac{1}{\sqrt{R^{2}+d^{2}}}\right]\)

An electric dipole is placed at an angle of \(30^{\circ}\) to a non-uniform electric field. The dipole will experience (A) a translational force only in the direction of the field. (B) a translational force only in a direction normal to the direction of the field. (C) a torque as well as a translational force. (D) a torque only.

Two spherical conductors \(A\) and \(B\) of radii \(1 \mathrm{~mm}\) and \(\mathrm{mm}\) are separated by a distance of \(5 \mathrm{~cm}\) and are uniformly charged. If the spheres are connected by conducting wire, then in equilibrium condition, the ratio of the magnitude of the electric fields at the surfaces of spheres \(A\) and \(B\) is (A) \(4: 1\) (B) \(1: 2\) (C) \(2: 1\) (D) \(1: 4\)

The potential at a point \(x\) (measured in \(\mu \mathrm{m}\) ) due to some charges situated on the \(x\)-axis is given by: \(V(x)=20 /\left(x^{2}-4\right)\) volt. The electric field \(E\) at \(x=4 \mu \mathrm{m}\) is given by (A) \(\frac{5}{3} \mathrm{~V} / \mu \mathrm{m}\) and in the \(-\) ve \(x\)-direction (B) \(\frac{5}{3} \mathrm{~V} / \mu \mathrm{m}\) and in the \(+\) ve \(x\)-direction (C) \(\frac{10}{9} \mathrm{~V} / \mu \mathrm{m}\) and in the \(-\) ve \(x\)-direction (D) \(\frac{10}{9} \mathrm{~V} / \mu \mathrm{m}\) and in the \(+\) ve \(x\)-direction

A thin spherical shell of radius \(R\) has charge \(Q\) spread uniformly over its surface. Which of the following graphs most closely represents the electric field \(E(r)\) produced by the shell in the range \(q 0 \leq r<\infty\), where \(r\) is the distance from the centre of the shell?

Let \(P(r)=\frac{Q}{\pi R^{4}} r\) be the charge density distribution for a solid sphere of radius \(R\) and total charge \(Q .\) For a point \(P\) inside the sphere at distance \(r_{i}\) from the centre of the sphere, the magnitude of electric field is (A) Zero (B) \(\frac{Q}{4 \pi \varepsilon_{0} r_{1}^{2}}\) (C) \(\frac{Q r_{1}^{2}}{4 \pi \varepsilon_{0} R^{4}}\) (D) \(\frac{Q r_{1}^{2}}{3 \pi \varepsilon_{0} R^{4}}\)

A charge \(Q\) is placed at the opposite corners of a square. A charge \(q\) is placed at each of the other two corners. If the net electrical force on \(Q\) is zero, then the \(\frac{Q}{q}\) equals (A) \(-2 \sqrt{2}\) (B) \(-1\) (C) 1 (D) \(-\frac{1}{\sqrt{2}}\)

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of \(30^{\circ}\) with each other. When suspended in a liquid of density \(8 \mathrm{~g} \mathrm{~cm}^{-3}\), the angle remains the same. If density of the material of the sphere is \(16 \mathrm{~g} \mathrm{~cm}^{-3}\), the dielectric constant of the liquid is (A) 4 (B) 3 (C) 2 (D) 1

Two identical charged spheres suspended from a common point by two massless strings of length \(I\) are initially a distance \(d(d<

In a uniformly charged sphere of total charge \(Q\) and radius \(R\), the electric field \(E\) is plotted as a function of distance from the centre. The graph which would correspond to the above will be

This question has Statement 1 and Statement \(2 . \mathrm{Of}\) the four choices given after the statements, choose the one that best describes the two statements. [2012] An insulating solid sphere of radius \(R\) has a uniformly positive charge density \(\rho .\) As a result of this uniform charge distribution, there is a finite value of electric potential at the centre of the sphere, at the surface of the sphere and also at a point out side the sphere. The electric potential at infinity is zero. Statement \(1:\) When a charge \(q\) is taken from the centre to the surface of the sphere, its potential energy changes by \(\frac{q \rho}{3 \varepsilon_{0}}\) Statement \(2:\) The electric field at a distance \(r(r

Two charges, each equal to \(q\), are kept at \(x=-a\) and \(x=a\) on the \(x\)-axis. A particle of mass \(m\) and charge \(q_{0}=\frac{q}{2}\) is placed at the origin. If charge \(q_{0}\) is given a small displacement \((y<

Two capacitors \(C_{1}\) and \(C_{2}\) are charged to \(120 \mathrm{~V}\) and \(200 \mathrm{~V}\), respectively. It is found that by connecting them together, the potential on each one can be made zero. Then (A) \(3 C_{1}=5 C_{2}\) (B) \(3 C_{1}+5 C_{2}=0\) (C) \(9 C_{1}=4 C_{2}\) (D) \(5 C_{1}=3 C_{2}\)

Assume that an electric field \(\vec{E}=30 x^{2} \hat{i}\) exists in space. Then the potential difference \(V_{A}-V_{O}\), where \(V_{O}\) is the potential at the origin and \(V_{A}\) the potential at \(x=2 \mathrm{~m}\) is (A) \(120 \mathrm{~J}\) (B) \(-120 \mathrm{~J}\) (C) \(-80 \mathrm{j}\) (D) \(80 \mathrm{~J}\)

A uniformly charged solid sphere of radius \(R\) has potential \(V_{0}\) (measured with respect to \(\infty\) ) on its surface. For this sphere, the equipotential surfaces with potential \(\frac{3 V_{0}}{2}, \frac{5 V_{0}}{4}, \frac{3 V_{0}}{4}\) and \(\frac{V_{0}}{4}\) have radius \(R_{1}, R_{2}\), \(R_{3}\), and \(R_{4}\), respectively. Then (A) \(R_{1} \neq 0\) and \(\left(R_{2}-R_{1}\right)>\left(R_{4}-R_{3}\right)\) (B) \(R_{1}=0\) and \(R_{2}<\left(R_{4}-R_{3}\right)\) (C) \(2 R\left(R_{4}-R_{3}\right)\)

A parallel plate capacitor is made of two circular plates separated by a distance of \(5 \mathrm{~mm}\) and with a dielectric constant of \(2.2\) between them. When the electric field in the dielectric is \(3 \times 10^{4} \mathrm{~V} / \mathrm{m}\), the charge density of the positive plate will be close to (A) \(6 \times 10^{-7} \mathrm{C} / \mathrm{m}^{2}\) (B) \(3 \times 10^{-7} \mathrm{C} / \mathrm{m}^{2}\) (C) \(3 \times 10^{4} \mathrm{C} / \mathrm{m}^{2}\) (D) \(6 \times 10^{4} \mathrm{C} / \mathrm{m}^{2}\)